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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1985 Swedish Mathematical Competition
1985 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
1
1
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(a-b)^2/8a < (a+b)/2 - \sqrt{ab} < (a-b)^2/8b for a > b > 0,
If
a
>
b
>
0
a > b > 0
a
>
b
>
0
, prove the inequality
(
a
−
b
)
2
8
a
<
a
+
b
2
−
a
b
<
(
a
−
b
)
2
8
b
.
\frac{(a-b)^2}{8a}< \frac{a+b}{2}- \sqrt{ab} < \frac{(a-b)^2}{8b}.
8
a
(
a
−
b
)
2
<
2
a
+
b
−
ab
<
8
b
(
a
−
b
)
2
.
6
1
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no of citizens wanted in X-wich, people and clubs related
X-wich has a vibrant club-life. For every pair of inhabitants there is exactly one club to which they both belong. For every pair of clubs there is exactly one person who is a member of both. No club has fewer than
3
3
3
members, and at least one club has
17
17
17
members. How many people live in X-wich?
5
1
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AB+BC+CA >= 2CO, triangle in coordinate system
In a rectangular coordinate system,
O
O
O
is the origin and
A
(
a
,
0
)
A(a,0)
A
(
a
,
0
)
,
B
(
0
,
b
)
B(0,b)
B
(
0
,
b
)
and
C
(
c
,
d
)
C(c,d)
C
(
c
,
d
)
the vertices of a triangle. Prove that
A
B
+
B
C
+
C
A
≥
2
C
O
AB+BC+CA \ge 2CO
A
B
+
BC
+
C
A
≥
2
CO
.
4
1
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p(x)+ p'(x)+ p''(x)+...+ p^{(n)}(x) >= 0 if p(x) >= 0
Let
p
(
x
)
p(x)
p
(
x
)
be a polynomial of degree
n
n
n
with real coefficients such that
p
(
x
)
≥
0
p(x) \ge 0
p
(
x
)
≥
0
for all
x
x
x
. Prove that
p
(
x
)
+
p
′
(
x
)
+
p
′
′
(
x
)
+
.
.
.
+
p
(
n
)
(
x
)
≥
0
p(x)+ p'(x)+ p''(x)+...+ p^{(n)}(x) \ge 0
p
(
x
)
+
p
′
(
x
)
+
p
′′
(
x
)
+
...
+
p
(
n
)
(
x
)
≥
0
.
3
1
Hide problems
DE = r wanted, inscribed isosceles ABC, equilateral BCD
Points
A
,
B
,
C
A,B,C
A
,
B
,
C
with
A
B
=
B
C
AB = BC
A
B
=
BC
are given on a circle with radius
r
r
r
, and
D
D
D
is a point inside the circle such that the triangle
B
C
D
BCD
BC
D
is equilateral. The line
A
D
AD
A
D
meets the circle again at
E
E
E
. Show that
D
E
=
r
DE = r
D
E
=
r
.
2
1
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least natural, first digit move last, new number is 7/2 times the original
Find the least natural number such that if the first digit (in the decimal system) is placed last, the new number is
7
/
2
7/2
7/2
times as large as the original number.